In complex analysis we have well known results about isolated singularities. Poles are characterized by ‘nice’ (rational) controlled growth around them and for essential singularities we have the Great Picard's Theorem.
Question: Is there a similar classification for branch points? I mean: a clear list with all possibilities and results that characterizes each case?
For example: if we compare $f(z)=\sqrt z$ and $g(z) = \sin(\ln(z))$, they have very different behavior, one has a well defined limit as we approach $z=0$ in any branch and the other has an accumulation point of zeros. Are there results that characterize the ‘fast oscillations’ of $g(z) = \sin(\ln(z))$ and the ‘calm’ behavior of $f(z)=\sqrt z$? (Maybe in an appropriate Riemann surface.)